Download Handout #5 AN INTRODUCTION TO VECTORS Prof. Moseley

Handout #5
AN INTRODUCT ION TO VECTO RS
Prof. Moseley
Traditionally the concept of a vector is introduced by physicists as a quantity such as force or velocity that
has magnitud e and direction. Vectors are then represented geome trically by directed line segments and are thought
of geometrically. Modern mathematical treatments introduce vectors algebraically as elements in a vector space.
As an introductory compromise we introduce two dimensional vectors algebraically and then examine the
correspondence between algebraic vectors and directed line segments in the plane. To define a vector space, we
need a set of vectors and a set of scalars. We also need to define the algebraic operations of vectors addition and
scalar multiplication. Since w e associate the analytic set R 2 = {(x,y): x,y 0 R} with the geom etric set of points in a
plane, we will use another notation, ú2, for the set of two dimensional algebra ic vectors.
SCALARS AND VECTORS. We define our set of scalars to b e the set of real numb ers R and o ur set of vectors to
be the set of ord ered pairs ú2 = {[x,y] T: x,y 0 R}. W e use the matrix notation [x,y] T (i.e.[x,y] transpose, see
Chapter 2-2) to indicate a column vector to save space.
W hen writing homework papers it is better to use column vectors explicitly. We write
= [x,y] T and refer to x and y as the compon ents of the vector
VECTOR ADDITION. We define the sum of the vectors
as the vector
+
.
= [x 1,y 1] T and
= [x 2,y2] T
= [x 1 + x 2, y1 + y2] T. For example, [2,1]T + [5,!2] T = [7,!1] T. Thus we add vectors
comp onent wise.
SCALAR MULTIPLICATION. We define scalar multiplication of a vector
by a scalar "0 R by "
= [x,y] T
= [" x, " y] T . For example, 3[2,!1] T = [6,!3] T. Thus we multiply each com ponent in
by the scalar ".
GEOMETRICAL INTERPRETATION. Reca ll that we associate the analytic set
R 2 = {(x,y): x,y 0 R} with the geom etric set of points in a plane. T empo rarily, we use
= (x,y) to deno te a po int in
R 2. We might say that the p oints in a p lane are a geo metric interpretation of R 2. We can establish a one-to-one
correspondence betwe en the analytic set R 2 and geometrical vectors (directed line segments). First consider only
directed line segments which are position vectors; that is, have their “tails” at the origin (i.e. at the p oint
and their heads at so me o ther po int, say, the point
Position vectors are said to be “based at
”. If
= (0,0)
2
= (x,y) in the plane R = {(x,y): x, y 0R}. D enote this set by G.
0R 2 is a point in the plane, then we let
denote the position
vector from the origin
to the point
. “Clearly” there exist a one-to-one correspondence between G and the set
R 2 of points in the plane; that is, we can readily identify exactly one vector in G with exactly one point in R 2. Now
“clearly” a one-to-one correspondence also exists betwe en the se t of points in R 2 and the set of algebraic vectors in
ú2. Hence a one-to-one correspondence exists between the set of algeb raic vectors ú2 and the set of ge ome tric
vectors G. In a traditional trea tment, vector addition and scalar multiplication are defined geometrically for directed
line segm ents in G. We must then prove that the geometric definition of the ad dition o f two directed line segm ents
using the parallelogram law corresponds to algebraic addition of the corresponding vectors in ú2. Similarly for
scalar multiplication. We say that we must prove that the two structures are isomo rphic. It is somewhat sim pler to
define vector addition and scalar multiplication algebraically on ú2 and think of G as a geometric interpretation of
the two dimensional vectors. One can then develop the theory of vectors without getting bogged down in the
geometric proofs to establish the isomorphism. We can extend this isomorphism to R 2. However, although we
readily accept adding geometric vectors in G and will accept with coaxing adding algebraic vectors in ú2 , we may
baulk at the idea of adding points in R 2. Howe ver, the distinction between these sets with structure really just
amounts to an interpretation of ordered pairs rather than an inherent difference in the mathematical objects being
considered. Hence from now o n, for any of these we will use the symb ol R 2 unless there is a ph ilosop hical reason to
make the distinction. Reviewing, we have
ú2 – G – R 2
where we have used the sym bol – to denote that there is an isomorp hism between these sets with structure.
MAGNITUDE AND DIRECTION. Normally we think of directed line segments as being “free” vectors; that is, we
identify any directed line segment in the plane with the directed line segment
in G which has the same magnitude (length) and direction. The magnitude of the algebraic
= [x,y] T 0 ú2 is defined to be **
vector
(1/**
** holds for all
**)
which is “clearly” the length of the directed line
. It is easy to show that the property (i.e. prove the theorem) **"
segment in G which is associated with
*"* **
** =
2
0 ú and all scalars ". It is also easy to show that if
… 0 the vector
has magnitude equal to one (i.e. is a unit vector). Examples of unit vectors are
T
[ 0, 1] . “Obviously” any nonzero vector
is a unit vector in the same direction as
T
= [x,y]
T
can be written as
. For example,
T
= **
**
= [ 3, 4] = 5 [3/5, 4/5]
T
where
** =
/**
** =
= [1, 0]T and
=
/**
**
=
where
2
= [3/5, 4/5] . To make vectors in ú more applicable to two dimensional geometry we can introduce the concept
of an equivalence relation and equivalence classes. We say that an arbitrary directed line segment in the p lane is
equivalent to a geometrical vector in G if it has the same direction and magnitude. The set of all directed line
segments equivalent to a given vector in G forms an equivalence class. Two directed line segments are related if
they are in the same equivalence classes. This relation is called an equivalence relation since all directed line
segments that are related can be tho ught of as being the sam e. Th e equ ivalence classes partition the set o f all
directed line segm ents into sets that are mutually exclusive whose union is all of the directed line segments.
VECTORS IN R 3. Having established an isomorphism between R 2, ú2, and G, we make no distinction in the future
and will usually use R 2 for the set of vectors in the plane, the set of points in the plane and the set of geometrical
vecto rs. The con text will exp lain what is meant. The same deve lopm ent can be done algebraically for vecto rs in
space (i.e. three dimensiona l vectors). There is a technical difference b etween the sets
R×R×R = {(x,y,z): x, y, z 0R},
R 2×R = {((x,y),z): x, y, z 0R}, and
R×R 2 = {(x,(y,z)): x, y, z 0R}
but they are isomorphic and we will usually co nsider them to be the sam e and deno te all three by R 3 and refer to this
set as the se t of ordered triples. The same analytic d evelo pme nt can b e done for 4 ,5,..., n to obtain the set R n of ndimensiona l vectors, but no “real” geom etrical interpretation is available. This d oes not dim inish the usefulness of
n-dimensional space since, for example, it plays a central role in the theory behind solving linear equations (m
equations in n unknowns) and the theory of any system which has n state variables. The impo rtant thing to
reme mbe r is that, although we may use geo metric language, we are doing algebra, not ge ome try.